A note on complex plane curve singularities up to diffeomorphism and their rigidity

We prove that, if two germs of plane curves $(C,0)$ and $(C^{\mathrm{\prime }},0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C^{\mathrm{\prime }}$ or to $\overline{C^{\mathrm{\prime }}}$. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component of zero-dimensional isosingular locus. Moreover, we provide a general overview of the different classifications of plane curve singularities.